You can apply for non-resident permits in many other states IF you have a permit in your home state. I have three non-res permits. Usually, they want to be sure your home state requirements are as "stiff" as theirs. That was easy for me, 'cuz my home state is RI.
Start at the Attorney Generals office. They may refer you to State Police, Dept of Public Safety, etc. Reciprocity is another deal.